Bring a large pot of water to a boil for the pasta.
In a deep large skillet, heat oil, 2 turns of the pan, over medium-high heat. Season the chicken with salt and pepper. Add the chicken to the pan and brown on all sides. Transfer to a plate. Melt in the butter, add the mushrooms and lightly brown, 5-6 minutes. Add the onion and garlic, season with salt and pepper, reduce the heat a bit and cook to soften the onion, 5-6 minutes.
Sprinkle in the flour and stir for 1 minute. Add the wine and cook until reduced by half. Add the stock and milk, and cook until the sauce is thick enough to coat the back of a spoon. Season with a few grates of nutmeg. Pull the spinach apart with your fingers as you add it to the pan and stir to heat it through. Adjust the seasonings.
Meanwhile, salt the boiling water and cook the pasta to al dente.
Stir the Parm into the sauce, return the chicken to the sauce and cook to heat through. Drain the pasta and toss it with the sauce and chicken.
Serve in shallow bowls and top with more Parm.